Physics Student Breaks Down Gymnastics Physics

[Narrator] Gymnastics is one of the most watched

Olympic events in America.

Millions of people tune in to watch elite athletes twist,

flip, and launch themselves in the air.

We'll take a look at three events

to see how these athletes master the physics

to pull off epic gymnastic feats.

Hi, I'm Emily.

I am a physics PhD student at Yale University.

In my past life, I was a level 10 gymnast.

Physics and gymnastics really go hand-in-hand,

and they're so, so interesting.

I'm always blown away by how much knowledge of physics

gymnasts and other athletes carry in their bodies.

It's really amazing to watch and to think about.

[upbeat music]

We're watching Leanne Wong

do a piece of an even bar routine.

First skill that she does is called a Van Leeuwen,

where she releases from the low bar, does a half turn,

and catches the high bar.

Then, she does her glide kip.

She casts up to a handstand.

She does two giants,

which are that motion where you go from a handstand

and then return back to a handstand to pick up speed

going into her dismount.

Two laid out flips with two twists.

This is a very difficult skill.

The gymnast technique has to be so good

in order for her to get that lift that she needs

to raise her center mass up enough

for her to grip the high bar.

This is a little bit more difficult

because it adds this half twist and she releases the bar

with one hand slightly before the other.

In doing so, she's applying a torque to the bar

and that's enabling her to get this half twist.

Something so cool about the bars

is you have a visible indication, a nice visual indication,

of how forces are at play because the bar bends

in accordance with those forces.

Something that is really interesting to compute

in a bar routine is just the acceleration

that she experiences at the bottom of her giant swings.

I'm making a lot of approximations

with all of these calculations.

There's a lot more going on than the simplified physics

that I'm doing,

but even so, it should start to give you

a bit of a Picture of what's going on

and why some of these moves are so challenging.

When she's at this point in her routine,

there are two forces acting on her right here.

Gravity, which is pointing downwards,

and compounding gravity.

She feels what's called a centrifugal force,

which is pulling her away from the bar

or pushing her downwards.

Centripetal acceleration is equal to V squared over R.

This V squared is for the center of mass of something

moving around an axis.

Velocity is distance over time.

And distance, in this case, if she's doing a giant,

is the circumference of a circle

traced out by her center of mass

as she goes completely around the bar.

The circumference of a circle

is two times PI times the radius,

and then this divided by the time it takes

for her to complete that one revolution.

And so, when we plug in for her distance to center of mass,

we'll call the radius about three feet

because she's about five feet,

and I think her full rotation is about one point,

the revolution is about 1.7 seconds.

In the end, we get velocity equals

roughly 3.4 meters per second.

Putting this back in our acceleration,

or centripetal acceleration, V squared over R,

plugging in numbers,

we get 12.5 meters per second squared,

which is roughly equivalent to

1.3 times gravitational acceleration.

But, as she's doing this swing,

it's not just this centripetal force that's acting on her.

There's also gravity as well.

So, the acceleration that she experiences

at the bottom of her swing is actually

a total equals the centripetal acceleration

plus acceleration due to gravity.

And I should say, this only holds

when she's at the very bottom of her swing.

And this would be 2.3 Gs of acceleration.

That's quite a lot.

It's, you can imagine hanging off a bar

and having something, an extra you holding onto you,

and you have to support that weight.

So, this is a lot of acceleration, and correspondingly,

a lot of force that Leanne's experiencing,

and she's just holding on with her hands.

You'll notice when a lot of gymnasts

are learning this skill, the most common place for them

to sort of peel off the bar or accidentally let go

is right at this point when they're moving the fastest

and also have these forces acting on them.

Oh my gosh, bars are my favorite.

I wish I had a better answer than they're just so fun.

[upbeat music]

Now, we're gonna take a look at Simone Biles on floor.

The tumbling pass we see from Simone is called The Biles,

named after her.

She does two flips in a laid out position

with a half turn right at the end.

It's incredibly difficult

and she was the first one to ever do it.

Part of what makes this skill so challenging

is that Simone is flipping in a laid out position

instead of a tuck.

There are physical reasons behind this,

and you can use physics equations

to sort of build a picture of why this is the case.

So, we can model Simone here

as flipping in her laid out position as a rod of length L.

So, L is the length of her body

spinning about some axis of rotation.

This will be the energy of a double layout,

will be proportional to the moment of inertia,

which is equal to, approximately for a rod,

1/12th ML squared.

For a double tuck, we're gonna approximate her

as a sphere when she's tucked.

And moment of inertia of a sphere is 2/5ths MR squared,

where R, if you're, if she's balled up in a sphere,

we're gonna call R approximately L over three.

If I tuck myself up,

the radius of my body is about a third.

If we wanna compare the energy of a double layout

to a double tuck, this is equal to 2/5ths,

and this will be L over three squared,

2/5ths ML squared over nine.

We can look at the ratio.

The layout over a tuck.

This is going to be equal to 1/12th over two over 45.

So, it's roughly two times as much energy

to complete a double layout than a double tuck.

And this is just accounting for the energy involved

and not even talking about how precisely

she needs to be able to place her body

in order to do this skill and remain so rigid,

and also get the height required and the rotation required.

We're looking at new Olympian, Jordan Chiles,

and in this tumbling pass,

she's doing a piked double Arabian with a half out.

The energy that she winds up with is built through her run,

it's built through these contact points

and how she manipulates her body to interact with the floor

and the springs.

She's running here, and then she contacts,

contacts, contacts, and releases to do her skill.

An interesting thing to look at with this tumbling pass

is how much energy is involved.

I'm gonna be making a lot of approximations

in this calculation.

It will not be exact.

It might even be off by a factor of two,

but it should still give you an idea

and some intuition for how hard what's going on actually is.

We'll need to know her mass, 55 kilograms;

her height, 1.524 meters.

That just corresponds to five foot even.

We'll also need to know her body's radius.

If you're looking directly at someone

from their stomach to their hips.

It's an approximation, that's gonna be 0.15 meters.

Acceleration due to gravity,

which is 9.81 meters per second squared.

This calculation is a little bit complicated

because it involves her moment of inertia,

which is the rotational analog of her mass.

And it's basically a description

of how her mass is organized

relative to the axis she's rotating around.

So, we're gonna approximate her twisting

as if she were a rod.

We're gonna talk about her in her piked position

when she flips as if she's a disc.

That's also gonna look like M disc radius squared over two.

For twisting and flipping, her full time in the air

is 1.125 seconds.

The distance is 1/2 AT squared.

Acceleration is just acceleration due to gravity.

So, this will be 1/2 GT squared.

This will tell you the distance from her highest point

to when she lands.

When she lands, she's at a velocity of zero.

And taking T is 1.125 seconds over two,

since the 1.125 was for her total arc

and this is only for her coming down.

The 1.52 meters is how much her center of mass

is raised above its landing point.

Height above ground

equals D plus center of mass height.

This is the D we're gonna be taking.

And now, we can compute her gravitational potential energy

at this piece.

So, E from gravity is mass times gravity times height,

which I get here to be 822 joules.

We can get the energy of her twist

is equal to 1/2 I omega squared.

So, I is this moment of inertia I was talking about.

Omega is her rotational velocity.

So, how fast she's spinning.

E flip will also be 1/2 I.

This is twist and I flip omega squared.

Twist is 10 joules

and this is 422 joules.

The total energy should be equal to the gravitational energy

plus her energy twisting plus her energy from flipping,

and you get 1,274 joules.

This number to put it in context is a lot of energy

for a person doing a jump.

If a person of this mass were to jump a foot and a half,

which is about standard for what American women can jump,

the E of a normal jump would be about 200 joules.

So, five to six times the energy of my jump

is what Jordan is doing here.

This calculation that I just did shows the energy involved

in Jordan's skill that she's doing here,

her piked double Arabian with a half out,

and showing how impressive it is

that there's so much energy involved.

[upbeat music]

In this clip we're looking at

now three-time Olympian Sam Mikulak

performing a Kasamatsu vault with one and a half twists.

The vault is just so much fun.

There's so many crazy physics things going on.

Look at that springboard.

Some of his momentum has transferred into the springboard.

He does only about a half rotation

before he's fully upright.

In the next half rotation of his flip,

he does two and a half twists.

And then, in the last half rotation,

he flails his arms out and he only does a half turn.

So, you can see how much of an impact it is

to have your arms in tight.

It's really hard to stick the landing on vaults,

especially coming from the heights,

and coming in with the power that these gymnasts have.

We can do some calculations to show

just what kinds of forces Sam is experiencing upon impact.

At this very moment, Sam is at his max downward velocity.

And then, when he lands, he'll come to a standstill.

He'll have zero translational velocity in the Y direction

and he'll only be moving sort of side to side

to get his balance.

So, if we can measure the time it takes

for him to decelerate and land.

So, his acceleration is his change in velocity

over that time.

This is his average acceleration,

it'll be higher at points and lower at points,

is 6.8 meters per second over 1/8th a second.

So, this is equal to

54.4 meters per second squared.

And in languages of gravity,

that's about 5.5 Gs.

That's what you experience on really fast roller coasters.

Watching these clips with a particular eye for the physics

has been really, really interesting

because it's sort of brought me back into trying to feel

how my body interacts with the equipment in different ways

and try to re-understand why that was happening.

So, looking through this lens of physics

has been particularly rewarding.

[upbeat music]